Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

World Journal of Condensed Matter Physics

Violation of Cluster Property in Heisenberg Antiferromagnet

Tomo Munehisa

Published Jan 1, 2018• 3 citations
Publisher

Abstract

There are concepts that are accepted in our daily life, but are not trivial in physics.One of them is the cluster property that means there exist no relations between two events which are sufficiently separated.In a paper recently published by the author, it has been pointed out that this cluster property violates in the correlation function of the spin operator for the spin 1/2 XXZ antiferromagnet on the square lattice.In this paper, we investigate the spin 1/2 Heisenberg antiferromagnet on the square lattice, which has SU(2) symmetry.In order to study the cluster property, we need to calculate the ground state accurately.For this purpose, we employ the effective model based on the magnetization of the sub-lattices.Then we can define the quasi-degenerate states to calculate the ground state.Including two kinds of interactions which break SU(2) symmetry into the Hamiltonian, we obtain the ground state quantitatively.We find that two kinds of spin correlation functions due to degenerate states are not zero when the lattice size is large but finite.The magnitude of one of them is the same as the one previously found in the XXZ antiferromagnet, while another one is much larger when the additional interaction is strong.We then conclude that in Heisenberg antiferromagnet correlation functions violate the cluster property and the magnitude of the violation qualitatively differs from the one in the XXZ antiferromagnet.

DOI
10.4236/wjcmp.2018.84015

Related papers

Violation of Cluster Property in Quantum Antiferromagnet

Tomo Munehisa

The cluster property is one of fundamental properties in physics.This property means that there are no relations between two events that are sufficiently separated.Because the cluster property is directly connected with entanglement in quantum field theory and in many-body systems, theoretical and experimental progress on entanglement stimulates us to study this property deeply.In this paper we investigate the cluster property in the spin 1/2 XXZ antiferromagnet on the square lattice with an explicitly symmetry breaking interaction of strength g.In this model spontaneous symmetry breaking occurs when the lattice size N is infinitely large.On the other hand, we have to make g zero in order to obtain quantities in the XXZ model with no symmetry breaking interaction.Since some results depend on the sequence of limit operations -N → ∞ and 0 g → , it is difficult to draw a clear conclusion in these limits.Therefore we study the model with finite g on the finite lattice, whose size N is supposed to be 10 20 , for our quantitative calculations.Then we can obtain the concrete ground state.In order to study the cluster property we calculate the spin correlation function.It is known that the function due to Nambu-Goldstone mode (gapless mode), which is calculated using linear spin wave theory, satisfies this property.In this paper we show that almost degenerate states also induce the spin correlation.We assume that the spin correlation function in measurements is a sum of the function due to Nambu-Goldstone mode and one due to these degenerate states.Then we examine whether the additional correlation function violates the cluster property or not.Our conclusion is that this function is finite at any distance, which means the violation of the cluster property, and it is of order of ( )for the case of extremely small g, this violation is the fine effect.Therefore the correlation function due to the degenerate states can be observed only when it is larger than the spin correlation function due to Nambu-Goldstone mode.We show that g required for this condition depends on the distance between positions of two spin operators.

World Journal of Condensed Matter Physics•Jan 1, 2018•4 citations

Quantum Curie-Weiss Magnet Induced by Violation of Cluster Property

Tomo Munehisa

There are some concepts that are accepted in our daily life but are not trivial in physics.One of them is the cluster property that means there exist no relations between two events which are sufficiently separated.In the works recently published by the author, the extensive and quantitative examination has been made about the violation of cluster property in the correlation function of the spin operator for the quantum spin system.These works have shown that, when we include the symmetry breaking interaction, the effect by the violation is proportional to the inverse of the system size.Therefore this effect is tinny since the system size is quite large.In order to find the effect due to the violation even when the size is large, we propose a new system where additional spins couple with the spin system on the square lattice, where the coupling constant between these systems being assumed to be small.Applying the perturbation theory, we obtain the effective Hamiltonian for the additional system.This Hamiltonian includes Curie-Weiss model that is induced by the violation of the cluster property.Then we find that this effective Hamiltonian has the factor which is the inverse of the system size.Since Curie-Weiss model, which is known to be exactly soluble, has to contain this factor so that the thermodynamical properties are well-defined, the essential factor for the Hamiltonian is determined by the coupling and the strength of the symmetry breaking interaction.Our conclusion is, therefore, that it is possible to observe the effect by the violation of the cluster property at the inverse temperature whose order is given by these parameters.

World Journal of Condensed Matter Physics•Jan 1, 2020•2 citations

Degenerate States in Nonlinear Sigma Model with U(1) Symmetry —For Study on Violation of Cluster Property

Tomo Munehisa

Entanglement in quantum theory is a concept that has confused many scientists.This concept implies that the cluster property, which means no relations between sufficiently separated two events, is non-trivial.In the works for some quantum spin systems, which have been recently published by the author, extensive and quantitative examinations were made about the violation of cluster property in the correlation function of the spin operator.The previous study of these quantum antiferromagnets showed that this violation is induced by the degenerate states in the systems where the continuous symmetry spontaneously breaks.Since this breaking is found in many materials such as the high temperature superconductors and the superfluidity, it is an important question whether we can observe the violation of the cluster property in them.As a step to answer this question we study a quantum nonlinear sigma model with U(1) symmetry in this paper.It is well known that this model, which has been derived as an effective model of the quantum spin systems, can also be applied to investigations of many materials.Notifying that the existence of the degenerate states is essential for the violation, we made numerical calculations in addition to theoretical arguments to find these states in the nonlinear sigma model.Then, successfully finding the degenerate states in the model, we came to a conclusion that there is a chance to observe the violation of cluster property in many materials to which the nonlinear sigma model applies.

World Journal of Condensed Matter Physics•Jan 1, 2021•1 citation

Degenerate States in Nonlinear Sigma Model with SU(2) Symmetry

Tomo Munehisa

Entanglement in quantum theory is a peculiar concept to scientists. With this concept we are forced to re-consider the cluster property which means that one event is irrelevant to another event when they are fully far away. In the recent works we showed that the quasi-degenerate states induce the violation of cluster property in antiferromagnets when the continuous symmetry breaks spontaneously. We expect that the violation of cluster property will be observed in other materials too, because the spontaneous symmetry breaking is found in many systems such as the high temperature superconductors and the superfluidity. In order to examine the cluster property for these materials, we studied a quantum nonlinear sigma model with U(1) symmetry in the previous work. There we showed that the model does have quasi-degenerate states. In this paper we study the quantum nonlinear sigma model with SU(2) symmetry. In our approach we first define the quantum system on the lattice and then adopt the representation where the kinetic term is diagonalized. Since we have no definition on the conjugate variable to the angle variable, we use the angular momentum operators instead for the kinetic term. In this representation we introduce the states with the fixed quantum numbers and carry out numerical calculations using quantum Monte Carlo methods and other methods. Through analytical and numerical studies, we conclude that the energy of the quasi-degenerate state is proportional to the squared total angular momentum as well as to the inverse of the lattice size.

World Journal of Condensed Matter Physics•Jan 1, 2023•0 citations

Study of the excited states of the quantum antiferromagnets

Mohammad G. Merdan

We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone $\Gamma(\mathbf q=0)$ and at the magnetic ordering wavevectors $\pm[\mathbf Q= (4\pi/3,0)]$, the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.

•Nov 10, 2013•0 citations

Ground state with nonzero spontaneous magnetization of the two-dimensional spin-1/2 Heisenberg antiferromagnet with frustration

Tôru Sakai, Hiroki Nakano

The S = 1/2 Heisenberg antiferromagnet on the two-dimensional pyramid lattice is studied by the numerical-diagonalization method. This lattice is obtained by the combination of the Lieb lattice and the square lattice. It is known that when interaction on the square lattice is increased from the ferrimagnetic limit of strong interaction on the Lieb lattice, this system shows gradual decrease and disappearance of spontaneous magnetization in the ground state. The present study treats the region near the case of the square-lattice antiferromagnet accompanied by isolated spins by numerical-diagonalization calculations of finite-size clusters with the maximum size of 39 sites. Our numerical results suggest the existence of a new phase with small but nonzero spontaneous magnetization between two zero-spontaneous-magnetization phases.

arXiv:1808.04997•AIP Advances•Aug 30, 2018•6 citations
There are concepts that are accepted in our daily life, but are not trivial in physics.One of them is the cluster property that means there exist no relations between two events which are sufficiently separated.In a paper recently published by the author, it has been pointed out that this cluster property violates in the correlation function of the spin operator for the spin 1/2 XXZ antiferromagnet on the square lattice.In this paper, we investigate the spin 1/2 Heisenberg antiferromagnet on the square lattice, which has SU(2) symmetry.In order to study the cluster property, we need to calculate the ground state accurately.For this purpose, we employ the effective model based on the magnetization of the sub-lattices.Then we can define the quasi-degenerate states to calculate the ground state.Including two kinds of interactions which break SU(2) symmetry into the Hamiltonian, we obtain the ground state quantitatively.We find that two kinds of spin correlation functions due to degenerate states are not zero when the lattice size is large but finite.The magnitude of one of them is the same as the one previously found in the XXZ antiferromagnet, while another one is much larger when the additional interaction is strong.We then conclude that in Heisenberg antiferromagnet correlation functions violate the cluster property and the magnitude of the violation qualitatively differs from the one in the XXZ antiferromagnet.
The cluster property is one of fundamental properties in physics.This property means that there are no relations between two events that are sufficiently separated.Because the cluster property is directly connected with entanglement in quantum field theory and in many-body systems, theoretical and experimental progress on entanglement stimulates us to study this property deeply.In this paper we investigate the cluster property in the spin 1/2 XXZ antiferromagnet on the square lattice with an explicitly symmetry breaking interaction of strength g.In this model spontaneous symmetry breaking occurs when the lattice size N is infinitely large.On the other hand, we have to make g zero in order to obtain quantities in the XXZ model with no symmetry breaking interaction.Since some results depend on the sequence of limit operations -N → ∞ and 0 g → , it is difficult to draw a clear conclusion in these limits.Therefore we study the model with finite g on the finite lattice, whose size N is supposed to be 10 20 , for our quantitative calculations.Then we can obtain the concrete ground state.In order to study the cluster property we calculate the spin correlation function.It is known that the function due to Nambu-Goldstone mode (gapless mode), which is calculated using linear spin wave theory, satisfies this property.In this paper we show that almost degenerate states also induce the spin correlation.We assume that the spin correlation function in measurements is a sum of the function due to Nambu-Goldstone mode and one due to these degenerate states.Then we examine whether the additional correlation function violates the cluster property or not.Our conclusion is that this function is finite at any distance, which means the violation of the cluster property, and it is of order of ( )for the case of extremely small g, this violation is the fine effect.Therefore the correlation function due to the degenerate states can be observed only when it is larger than the spin correlation function due to Nambu-Goldstone mode.We show that g required for this condition depends on the distance between positions of two spin operators.
There are some concepts that are accepted in our daily life but are not trivial in physics.One of them is the cluster property that means there exist no relations between two events which are sufficiently separated.In the works recently published by the author, the extensive and quantitative examination has been made about the violation of cluster property in the correlation function of the spin operator for the quantum spin system.These works have shown that, when we include the symmetry breaking interaction, the effect by the violation is proportional to the inverse of the system size.Therefore this effect is tinny since the system size is quite large.In order to find the effect due to the violation even when the size is large, we propose a new system where additional spins couple with the spin system on the square lattice, where the coupling constant between these systems being assumed to be small.Applying the perturbation theory, we obtain the effective Hamiltonian for the additional system.This Hamiltonian includes Curie-Weiss model that is induced by the violation of the cluster property.Then we find that this effective Hamiltonian has the factor which is the inverse of the system size.Since Curie-Weiss model, which is known to be exactly soluble, has to contain this factor so that the thermodynamical properties are well-defined, the essential factor for the Hamiltonian is determined by the coupling and the strength of the symmetry breaking interaction.Our conclusion is, therefore, that it is possible to observe the effect by the violation of the cluster property at the inverse temperature whose order is given by these parameters.
Entanglement in quantum theory is a concept that has confused many scientists.This concept implies that the cluster property, which means no relations between sufficiently separated two events, is non-trivial.In the works for some quantum spin systems, which have been recently published by the author, extensive and quantitative examinations were made about the violation of cluster property in the correlation function of the spin operator.The previous study of these quantum antiferromagnets showed that this violation is induced by the degenerate states in the systems where the continuous symmetry spontaneously breaks.Since this breaking is found in many materials such as the high temperature superconductors and the superfluidity, it is an important question whether we can observe the violation of the cluster property in them.As a step to answer this question we study a quantum nonlinear sigma model with U(1) symmetry in this paper.It is well known that this model, which has been derived as an effective model of the quantum spin systems, can also be applied to investigations of many materials.Notifying that the existence of the degenerate states is essential for the violation, we made numerical calculations in addition to theoretical arguments to find these states in the nonlinear sigma model.Then, successfully finding the degenerate states in the model, we came to a conclusion that there is a chance to observe the violation of cluster property in many materials to which the nonlinear sigma model applies.
Entanglement in quantum theory is a peculiar concept to scientists. With this concept we are forced to re-consider the cluster property which means that one event is irrelevant to another event when they are fully far away. In the recent works we showed that the quasi-degenerate states induce the violation of cluster property in antiferromagnets when the continuous symmetry breaks spontaneously. We expect that the violation of cluster property will be observed in other materials too, because the spontaneous symmetry breaking is found in many systems such as the high temperature superconductors and the superfluidity. In order to examine the cluster property for these materials, we studied a quantum nonlinear sigma model with U(1) symmetry in the previous work. There we showed that the model does have quasi-degenerate states. In this paper we study the quantum nonlinear sigma model with SU(2) symmetry. In our approach we first define the quantum system on the lattice and then adopt the representation where the kinetic term is diagonalized. Since we have no definition on the conjugate variable to the angle variable, we use the angular momentum operators instead for the kinetic term. In this representation we introduce the states with the fixed quantum numbers and carry out numerical calculations using quantum Monte Carlo methods and other methods. Through analytical and numerical studies, we conclude that the energy of the quasi-degenerate state is proportional to the squared total angular momentum as well as to the inverse of the lattice size.
We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone Γ(q=0)\Gamma(\mathbf q=0)Γ(q=0) and at the magnetic ordering wavevectors ±[Q=(4π/3,0)]\pm[\mathbf Q= (4\pi/3,0)]±[Q=(4π/3,0)], the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.
The S = 1/2 Heisenberg antiferromagnet on the two-dimensional pyramid lattice is studied by the numerical-diagonalization method. This lattice is obtained by the combination of the Lieb lattice and the square lattice. It is known that when interaction on the square lattice is increased from the ferrimagnetic limit of strong interaction on the Lieb lattice, this system shows gradual decrease and disappearance of spontaneous magnetization in the ground state. The present study treats the region near the case of the square-lattice antiferromagnet accompanied by isolated spins by numerical-diagonalization calculations of finite-size clusters with the maximum size of 39 sites. Our numerical results suggest the existence of a new phase with small but nonzero spontaneous magnetization between two zero-spontaneous-magnetization phases.